Problem 20 - Algebra for College Students

Question

For Problems \(13-26\), compute \(A B\) and \(B A\). $$ A=\left[\begin{array}{ll} -2 & 3 \\ -1 & 7 \end{array}\right], \quad B=\left[\begin{array}{ll} -1 & -3 \\ -5 & -7 \end{array}\right] $$

Step-by-Step Solution

Answer

The product matrices are \(AB = \left[\begin{array}{cc} -13 & -15 \\ -34 & -46 \end{array}\right]\) and \(BA = \left[\begin{array}{cc} 5 & -24 \\ 17 & -64 \end{array}\right]\). They are not equal, showing non-commutative property.

Step 1: Understand Matrix Multiplication Rules

Matrix multiplication involves multiplying the rows of the first matrix by the columns of the second matrix. For two matrices \(A\) and \(B\), to find the element \(c_{ij}\) of the resulting matrix \(C = AB\), we calculate \(c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j}\) and so on, using each element from the row of \(A\) and column of \(B\).

Step 2: Multiply Matrices A and B

Calculate \(AB\):- First row, first column: \((-2)(-1) + (3)(-5) = 2 - 15 = -13\).- First row, second column: \((-2)(-3) + (3)(-7) = 6 - 21 = -15\).- Second row, first column: \((-1)(-1) + (7)(-5) = 1 - 35 = -34\).- Second row, second column: \((-1)(-3) + (7)(-7) = 3 - 49 = -46\).Thus, \(AB = \left[\begin{array}{cc} -13 & -15 \ -34 & -46 \end{array}\right]\).

Step 3: Multiply Matrices B and A

Calculate \(BA\):- First row, first column: \((-1)(-2) + (-3)(-1) = 2 + 3 = 5\).- First row, second column: \((-1)(3) + (-3)(7) = -3 - 21 = -24\).- Second row, first column: \((-5)(-2) + (-7)(-1) = 10 + 7 = 17\).- Second row, second column: \((-5)(3) + (-7)(7) = -15 - 49 = -64\).Thus, \(BA = \left[\begin{array}{cc} 5 & -24 \ 17 & -64 \end{array}\right]\).

Step 4: Compare Non-commutative Results

The computed matrices \(AB\) and \(BA\) are not equal. This demonstrates that matrix multiplication is not commutative, meaning \(AB eq BA\).

Key Concepts

Non-commutative Property

Matrix multiplication is a foundational concept in linear algebra that exhibits the non-commutative property. This property means that when multiplying two matrices, the order of multiplication matters. In other words, multiplying matrix \(A\) by matrix \(B\) may produce a different result than multiplying \(B\) by \(A\). This is unlike scalar multiplication, which is commutative and allows for changing the order of operands without affecting the outcome.

In the example provided, multiplying matrices \(A\) and \(B\) yields two distinct results: \(AB\) produces the matrix \[\begin{bmatrix} -13 & -15 \ -34 & -46 \end{bmatrix}\] while \(BA\) results in the matrix \[\begin{bmatrix} 5 & -24 \ 17 & -64 \end{bmatrix}\]. The clear difference between these resulting matrices illustrates that matrix multiplication is inherently non-commutative. In practice, recognizing this property is crucial when solving equations or performing operations involving matrices, as assuming commutativity can lead to incorrect conclusions or solutions.

• Remember: \(AB eq BA\)
• Order of multiplication impacts the result
• Essential for correctly performing matrix operations

Matrix Operations

Matrix operations encompass a variety of mathematical procedures like addition, subtraction, multiplication, and finding the inverse of matrices. These operations are fundamental in fields like computer graphics, physics, engineering, and more. Each operation has its own set of rules, with matrix multiplication being one of the most prominent given its application in transformations and solving systems of equations.

When multiplying matrices, the number of columns in the first matrix must match the number of rows in the second matrix. This is essential for the operation to be defined. The product matrix's dimensions are determined by the number of rows of the first matrix and the number of columns of the second matrix.

• Multiplication requires compatible dimensions
• Results stored in a new matrix of defined dimensions
• Different from scalar multiplication: involves row-column interactions

Thus, in the example with matrices \(A\) and \(B\), each element in the resulting matrix is calculated as a sum of products of corresponding elements from the rows of \(A\) and columns of \(B\). This interaction is what makes matrix multiplication multifaceted and vastly applicable.

Algebra

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating these symbols to solve equations and understand abstract structures. Matrices are algebraic structures used to represent data and transformations.

In algebra, understanding how matrices work allows us to solve complex systems of linear equations and perform linear transformations efficiently. The algebraic manipulation of matrices involves applying rules such as distribution in addition and sticking to the unique rules for multiplication. The distinction between solving equations with numbers versus structures like matrices requires a solid comprehension of both algebraic principles and properties.

• Symbols and rules are foundational
• Matrix algebra helps in solving linear systems
• Combines elements of pure algebra with practical applications

The impact of algebra in matrix operations is significant, extending to the ability to model and solve real-world problems. By applying algebraic rules, one can analyze data, simulate scenarios, and predict outcomes efficiently, always considering the properties that govern matrices.